The fox hunting situation in the UK is... colourful.
Foxes are a native species here but they are deemed a pest for the same reason as you mentioned. Their numbers are high and they wreck havoc on the agriculture industry.

In the UK the ruling/elite class like to ride around the countryside on horse back with packs of hounds until they flush out a fox and chase it to exhaustion and let the dogs rip it to pieces.

The above form of hunting is technically illegal. They are only allowed to trail hunt. Which means someone goes out before the hunt and drags a bag that is scented for the hounds to follow, seems harmless enough on paper, and the police allow them to do that. However while they're chasing the scent, obviously they're going to come across foxes anyway, and they end up getting shredded. But it's legal because "they didn't mean to".

They should be culled humanely as you do. But our ruling class are cruel and terrible humans.